Let $\Gamma$ be a finitely generated group, $G_\mathbb{ C}$ be a classical complex group and $G_\mathbb{ R}$ a real form of $G_\mathbb{ C}$. We propose a definition of the $G_\mathbb{ R}$-character variety of $\Gamma$ as a subset $\mathcal{ X}_{ G_\mathbb{ R}}(\Gamma)$ of the $G_\mathbb{ C}$-character variety $\mathcal{ X}_{ G_\mathbb{ C}}(\Gamma)$. We prove that these subsets cover the set of irreducible $G_\mathbb{ C}$-characters fixed by an anti-holomorphic involution $\Phi$ of $\mathcal{ X}_{ G_\mathbb{ C}}(\Gamma)$. Whenever $G_\mathbb{ R}$ is compact, we prove that $\mathcal{ X}_{ G_\mathbb{ R}}(\Gamma)$ is homeomorphic to the topological quotient $\mathrm{ Hom}(\Gamma,G_\mathbb{ R})/G_\mathbb{ R}$. Finally, we identify the reducible points of $\mathcal{ X}_{ \mathrm{ GL}(n,\mathbb{ C})}(\Gamma)$ fixed by an anti-holomorphic involution $\Phi$ as coming from direct sums of representations with values in real groups.
Categories: math.GT math.GR